2 196 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Introduction Spectral method for olving PDE on unbounded domain can be eentially claified into four approache: (i) Domain truncation: truncate unbounded domain to bounded domain and olve the PDE on bounded domain upplemented with artificial or tranparent boundary condition (ee, e.g., [17, 21, 22, 25, 44, 51]); (ii) Approximation by claical orthogonal ytem on unbounded domain, e.g., Laguerre or Hermite polynomial/function (ee, e.g., [7, 14, 2, 3, 31, 36, 43, 47]); (iii) Approximation by other, non-claical orthogonal ytem (ee, e.g., [14]), or by mapped orthogonal ytem, e.g., image of claical Jacobi polynomial though a uitable mapping (ee, e.g. [32, 34, 35, 54]); (iv) Mapping: map unbounded domain to bounded domain and ue tandard pectral method to olve the mapped PDE in the bounded domain (ee, e.g., [9 12, 15, 24, 26]). Boyd provided in [11] an excellent review on general propertie and practical implementation for many of thee approache. In general, the domain truncation approach i only a viable option for problem with rapidly (exponentially) decaying olution or when accurate non-reflecting or exact boundary condition are available at the truncated boundary. On the other hand, with proper choice of mapping and/or caling parameter, the other three approache can all be effectively applied to a variety of problem with rapid or low decaying (or even growing) olution. Since there i a vat literature on domain truncation, particularly for Helmholtz equation and Maxwell equation for cattering problem and the analyi involved i very different from the other three approache, the domain truncation approach will not be addreed in thi paper. We note that the lat two approache are mathematically equivalent (ee Section for more detail) but their computational implementation are different. More preciely, the lat approach involve olving the mapped PDE (which are often cumberome to deal with) uing claical Jacobi polynomial while the third approach olve the original PDE uing the mapped Jacobi polynomial. The main advantage of the lat approach i that it can be implemented and analyzed uing tandard procedure and approximation reult, but it main diadvantage i that the tranformed equation i uually very complicated which, in many cae, make it implementation and analyi unuually cumberome. On the other hand, we work on the original PDE in the third approach and approximate it olution by uing a new family of orthogonal function which are image of claical Jacobi polynomial under a uitable mapping. The analyi of thi approach will require approximation reult by the new family of orthogonal function. The main advantage i that once thee approximation reult are etablihed, they can be directly applied to a large cla of problem. Thu, we hall mainly concentrate on the econd and third approache, and provide a general framework for the analyi of thee pectral method. While pectral method have been ued for olving PDE on unbounded domain

3 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp for over thirty year, and there have been everal iolated effort in the early year on the error analyi of thee method (ee, e.g. [6, 7, 15, 2, 23, 42]), it i only in the lat ten year or o that the baic approximation propertie of thee orthogonal ytem, and their application to PDE, were ytematically tudied (cf. [13] for a brief account). However, many of thee analye ue different approache and involve complicated Sobolev pace, making it hard for non-expert to extract ueful information from thee error etimate and to carry out error analyi for their application. The main purpoe of thi paper are three fold: (i) to preent a unified framework, for the analyi of mapped Jacobi, Laguerre and Hermite pectral method, which lead to more concie reult (than thoe appeared in the literature) and optimal approximation reult in mot ituation; (ii) to make a detailed comparion on the convergence rate of different method for everal typical olution; and (iii) to provide a brief (by no mean complete) review on ome of the recent work for the analyi and application of pectral method in unbounded domain. Thi paper i organized a follow. In the next ection, we conider the mapped pectral method and preent a unified framework to tudy their convergence propertie. In Section 3, we conider the approximation by the (generalized) Laguerre polynomial/function, and Section 4 i devoted to the approximation by the Hermite polynomial/function. Thee three ection are preented with a unified tyle and encompa mot of the important approximation reult on thee orthogonal ytem developed in the lat few year. In Section 5, we provide ome implementation detail and compare the performance of different method with two typical example. In Section 6, we dicu variou extenion and other iue related to the application of thee pectral method. We end thi paper with a few concluding remark. We now introduce ome notation. Let ω(x) be a certain weight function in Ω:=(a,b), where a or b could be infinite. We hall ue the weighted Sobolev pace H r ω(ω) (r =,1,2, ), whoe inner product, norm and emi-norm are denoted by (, ) r,ω, r,ω and r,ω, repectively. For real r>, we define the pace H r ω(ω) by pace interpolation. In particular, the norm and inner product of L 2 ω(ω) = H ω(ω) are denoted by ω and (, ) ω, repectively. The ubcript ω will be omitted from the notation in cae of ω 1. For notational convenience, we denote k x=d k /dx k, k 1, and for any nonnegative integer, let P be the et of all algebraic polynomial of degree. We denote by c a generic poitive contant independent of any function and, and ue the expreion A B to mean that there exit a generic poitive contant c uch that A cb. 2 Mapped Jacobi method A common and effective trategy in dealing with an unbounded domain i to ue a uitable mapping that tranform an infinite domain to a finite domain. Then, image of claical orthogonal polynomial under the invere mapping will form a et of orthogonal bai function which can be ued to approximate olution of PDE in the infinite domain.

9 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Remark 2.1. It hould be pointed out that under the above general etting, the approximation reult on the higher-order proection, uch a the H 1 (Λ) orthogonal proec- ω α,β tion π 1,α,β, : H1 ω α,β (Λ) V α,β,, can be etablihed by uing the exiting Jacobi approximation reult (ee, e.g., [38]) and a imilar argument a above. In particular, applying the above reult with α=β=, 1/2 to the algebraic mapping (2.12) and (2.15) lead to more concie and in ome cae improved, Chebyhev and Legendre rational approximation reult which were developed eparately in [34, 35, 39, 54]. The error etimate in the above theorem look very imilar to the uual pectral error etimate in a finite interval (cf. Lemma 2.1). Firt of all, it i clear from the above theorem that the proection error converge fater than any algebraic rate if a function decay exponentially fat at infinity. For a function with ingularitie inide the domain, the above theorem and Lemma 2.1 lead to the ame order of convergence, auming that the function decay ufficiently fat at infinity. However, for a given mooth function, they may lead to very different convergence rate due to the difference in the norm ued to meaure the regularity. We now determine the convergence rate for three et of function with typical decay propertie: Set 1. Exponential decay with ocillation at infinity u(x)=inkxe x for x (, ) or u(x)=inkxe x2 for x (, ). (2.31) Set 2. Algebraic decay without ocillation at infinity u(x)=(1+x) h for x (, ) or u(x)=(1+x 2 ) h for x (, ). (2.32) Set 3. Algebraic decay with ocillation at infinity u(x)= inkx inkx for x (, ) or u(x)= (1+x) h (1+x 2 for x (, ). (2.33) ) h Conider firt the mapping (2.15). Then, D x = ( dy ) 1 d dx dx = (x+)2 d 2 dx, ( 2 ) k ( 2x ) l 2 ωk,l (x)= x+ x+ (x+) 2. Hence, for u(x)=(1+x) h, it can be eaily checked that Dx mu ω α+m,β+m < if m<2h+α+1, which implie that u π α,β u ω α,β (2h+α+1) (u(x)=(1+x) h ). (2.34)

12 26 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp (a) Ditribution of x (, ) n=2 n=16 n=8 n= x (c) Ditribution of x (, ) n=8 n=12 n=16 n=2 (b) Effect of caling factor =2. =.5 =.1 =1. = x (d) Effect of caling factor =2., m=13 =1.5, m=15 =1., m=17 =.5, m=2 =.1, m= x x Figure 1: (a) Hermite-Gau point ( ) v. mapped Legendre-Gau point uing the algebraic map (2.12) with =1 ( ) for variou n; (b) Mapped Legendre-Gau point with n=16 and variou caling factor ; (c) Laguerre-Gau-Radau point ( ) v. mapped Legendre-Gau-Radau point uing the algebraic map (2.15) with =1 ( ) for variou n; (d) Mapped Legendre-Gau-Radau point with n =32 and variou caling factor (m i the number of point in the ubinterval [,1)). In Fig. 1, we plot ample grid ditribution for different caling factor with variou number of node for the mapped Legendre Gau (or Gau-Radau) point (ee the caption for detail). A comparion with Hermite-Gau point i alo preented in Fig. 1(a). We notice that the mapped Legendre-Gau point are more clutered near the origin and pread further, while the Hermite-Gau point are more evenly ditributed. It hould be oberved that the ditribution of mapped Legendre-Gau point i more favorable ince a much larger effective interval i covered. However, it can be hown that in both cae,

13 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp the mallet ditance between neighboring point i O( 1 ), a oppoed to O( 2 ) for Jacobi-Gau type node in a finite interval. A comparion of mapped Legendre and Laguerre Gau-Radau node i hown in Fig. 1(c). The mapped Legendre-Gau-Radau point are much more clutered near the origin, and one can check that the mallet ditance between neighboring point i O( 2 ), a oppoed to O( 1 ) for the Laguerre Gau-Radau node. Hence, the ditribution of mapped Legendre-Gau-Radau point i more favorable a far a reolution/accuracy i concerned but it will lead to a more retrictive CFL condition if explicit cheme are ued for time-dependent problem. 2.5 umerical method uing mapped Jacobi polynomial A generic example Conider the model equation γu x (a(x) x u)= f, x Λ=(,+ ), γ, (2.46) with uitable decay condition at ± which will depend on the weight function in the weighted variational formulation. For a given mapping x=g(y;) with x Λ and y ( 1,1), we recall that the mapped Jacobi polynomial are mutually orthogonal in L 2 ω α,β for (2.46) i to find u V α,β, uch that γ(u,v ) ω α,β + ( a(x) x u, x (v ω α,β ) ) =(I α,β (Λ). Hence, the mapped Jacobi method, f,v ) ω α,β, v V α,β,. (2.47) Let u now conider the econd approach decribed in the introduction. Here, Eq. (2.46) i firt tranformed into γu 1 ( a(g(y;)) ) g (y;) y g (y;) yu = F, (2.48) where U (y)=u(g(y;)) and F (y)= f(g(y;)). Then, let ˆω α,β (y)=ω α,β (y)g (y;), the Jacobi pectral method for (2.48) i to find ũ P uch that ( a(g(y;)) ) γ(ũ,ṽ ) ω α,β+ g (y;) yũ, y (ṽ ˆω α,β ) =(I α,β F,ṽ ) ω α,β, ṽ P. (2.49) One can verify eaily that ũ (y)=u (g(y;)). Hence, the two approache are mathematically equivalent. We remark that the formulation (2.49) i in general more difficult to analyze due to the ingular nature of g (y;), while the analyi for the formulation (2.47) become tandard once we etablih the baic approximation propertie of the mapped Jacobi polynomial.

15 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Let u denote X ={u V α,β, :u()=}. We can then define the Galerkin approximation of (2.52) by the mapped Jacobi polynomial a follow: For f L 2 ω(λ) C( Λ), find u X uch that a ω (u,v )=(I α,β, f,v ) ω, v X. (2.53) Unlike the tandard pectral method in a finite domain, the well-poedne of (2.52) and of (2.53) i not guaranteed for all cae with γ. A general reult for the wellpoedne of an abtract equation of the form (2.52) i etablihed in [49]. For the reader convenience, we recall thi reult below (cf. Lemma 2.3 in [49]): Lemma 2.3. We aume that d 1 =max x Λ ω 1 (x) x ω(x), d 2 =max ω 1 (x) 2 x ω(x) x Λ are finite and that u 2 (x)ω (x) x Λ = for u H 1,ω (Λ). Then, for any u,v H1 ω(λ), we have that a ω (u,v) (d 1 +1) u 1,ω v 1,ω +γ u ω v ω, (2.54) and for any v H 1,ω (Λ), a (ν) ω (v,v) v 2 1,ω +(γ d 2/2) v 2 ω. (2.55) Remark 2.2. The inequality (2.55) i derived under a general framework. For a pecific problem, the contant γ d 2 /2 can often be replaced by a larger contant. Thank to the above lemma, it i then traightforward to prove the following general reult: Theorem 2.3. Aume that the condition of Lemma (2.3) are atified and γ d 2 /2>. Then the problem (2.52) (rep. (2.53)) admit a unique olution. Furthermore, we have the error etimate: u u 1,ω inf v X u v 1,ω + f I α,β, f ω. (2.56) Remark 2.3. With a change of variable x to x/c (c>) for Eq. (2.46), the retriction on γ can be relaxed to γ>. Hence, given a mapping and a pair of Jacobi parameter (α,β), we ut need to compute upper bound for d 1 and d 2, verify that the condition of Theorem 2.3 are atified, and apply the approximation reult in Theorem 2.1 and 2.2 to (2.56) to get the deired error etimate. Conider for example the mapped Legendre method for (2.52) with the mapping (2.15). It can be hown that for thi mapping, we have d 1 2 and d 2 6. Applying Theorem 2.1 and 2.2 to (2.56) with (α,β)=(,) lead to the following reult:

Harmonic Ocillation / Complex Number Overview and Motivation: Probably the ingle mot important problem in all of phyic i the imple harmonic ocillator. It can be tudied claically or uantum mechanically,

Baic Quantum Mechanic in Coorinate, Momentum an Phae Space Frank Rioux Department of Chemitry College of St. Beneict St. Johnʹ Univerity The purpoe of thi paper i to ue calculation on the harmonic ocillator

Unit 11 Uing Linear Regreion to Decribe Relationhip Objective: To obtain and interpret the lope and intercept of the leat quare line for predicting a quantitative repone variable from a quantitative explanatory

Control of Wirele Network with Flow Level Dynamic under Contant Time Scheduling Long Le and Ravi R. Mazumdar Department of Electrical and Computer Engineering Univerity of Waterloo,Waterloo, ON, Canada

SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol 4 No June Mixed Method of Model Reduction for Uncertain Sytem N Selvaganean Abtract: A mixed method for reducing a higher order uncertain ytem to a table reduced

Solution of the Heat Equation for tranient conduction by LaPlace Tranform Thi notebook ha been written in Mathematica by Mark J. McCready Profeor and Chair of Chemical Engineering Univerity of Notre Dame

Chapter 2 Motion in One Dimenion 2.1 The Important Stuff 2.1.1 Poition, Time and Diplacement We begin our tudy of motion by conidering object which are very mall in comparion to the ize of their movement

TEHNIA NOTE 3 The quartz crytal model and it frequencie. Introduction The region between and i a region of poitive In thi note, we preent ome of the baic electrical propertie of quartz crytal. In particular,

Key to learning in pecific ubject area of engineering education an example from electrical engineering Anna-Karin Cartenen,, and Jonte Bernhard, School of Engineering, Jönköping Univerity, S- Jönköping,

22 Differential Equation Intructor: Petronela Radu November 8 25 Solution to Sample Problem for Tet 3 For each of the linear ytem below find an interval in which the general olution i defined (a) x = x

Inference When Comparing Two Mean Dr. Tom Ilvento FREC 48 Thu far We have made an inference from a ingle ample mean and proportion to a population, uing The ample mean (or proportion) The ample tandard

Heat tranfer to or from a fluid flowing through a tube R. Shankar Subramanian A common ituation encountered by the chemical engineer i heat tranfer to fluid flowing through a tube. Thi can occur in heat

Turbulent Mixing and Chemical Reaction in Stirred Tank André Bakker Julian B. Faano Blend time and chemical product ditribution in turbulent agitated veel can be predicted with the aid of Computational

6 MODULE 2. FUNDAMENTALS OF ALGEBRA 2b Order of Operation Simplifying Algebraic Expreion Recall the commutative and aociative propertie of multiplication. The Commutative Property of Multiplication. If

Introduction to the article Degree of Freedom. The article by Walker, H. W. Degree of Freedom. Journal of Educational Pychology. 3(4) (940) 53-69, wa trancribed from the original by Chri Olen, George Wahington

Section 3.4 Pre-Activity Preparation Quadrilateral Intereting geometric hape and pattern are all around u when we tart looking for them. Examine a row of fencing or the tiling deign at the wimming pool.

Chapter Stoc and Their Valuation ANSWERS TO EN-OF-CHAPTER QUESTIONS - a. A proxy i a document giving one peron the authority to act for another, typically the power to vote hare of common toc. If earning

International Journal of Advanced Technology & Engineering Reearch (IJATER) REDUCTION OF TOTAL SUPPLY CHAIN CYCLE TIME IN INTERNAL BUSINESS PROCESS OF REAMER USING DOE AND Abtract TAGUCHI METHODOLOGY Mr.

Click Here for Full Article GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L16804, doi:10.1029/2009gl039667, 2009 A model for the relationhip between tropical precipitation and column water vapor Caroline J. Muller,

Complex Stock Trading Strategy Baed on Particle Swarm Optimization Fei Wang, Philip L.H. Yu and David W. Cheung Abtract Trading rule have been utilized in the tock market to make profit for more than a

Chapter 4: Mean-Variance Analyi Modern portfolio theory identifie two apect of the invetment problem. Firt, an invetor will want to maximize the expected rate of return on the portfolio. Second, an invetor

TIME SERIES ANALYSIS AND TRENDS BY USING SPSS PROGRAMME RADMILA KOCURKOVÁ Sileian Univerity in Opava School of Buine Adminitration in Karviná Department of Mathematical Method in Economic Czech Republic

Tranient turbulent flow in a pipe M. S. Ghidaoui A. A. Kolyhkin Rémi Vaillancourt CRM-3176 January 25 Thi work wa upported in part by the Latvian Council of Science, project 4.1239, the Natural Science

EXISTENCE AND NON-EXISTENCE OF SOLUTIONS TO ELLIPTIC EQUATIONS WITH A GENERAL CONVECTION TERM SALOMÓN ALARCÓN, JORGE GARCÍA-MELIÁN AND ALEXANDER QUAAS Abtract. In thi paper we conider the nonlinear elliptic

7 THE SKOLIAD CORNER No. 51 R.E. Woodrow In the November 000 number of the Corner we gave the problem of the Final Round of the Britih Columbia College Junior High School Mathematic Contet, May 5, 000.

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

MA 408 Homework 4 Remark 0.1. When dealing with coordinate function, I continually ue the expreion ditance preerving throughout. Thi mean that you can calculate the ditance in the geometry P Q or you can

DIHEDRAL GROUPS KEITH CONRAD 1. Introduction For n 3, the dihedral group D n i defined a the rigid motion 1 of the plane preerving a regular n-gon, with the operation being compoition. Thee polygon for

Section 8. 8. ield Criteria in Three Dimenional Platicity The quetion now arie: a material yield at a tre level in a uniaxial tenion tet, but when doe it yield when ubjected to a complex three-dimenional

RISK MANAGEMENT POLICY The practice of foreign exchange (FX) rik management i an area thrut into the potlight due to the market volatility that ha prevailed for ome time. A a conequence, many corporation

Allen M. Potehman Univerity of Illinoi at Urbana-Champaign Unuual Option Market Activity and the Terrorit Attack of September 11, 2001* I. Introduction In the aftermath of the terrorit attack on the World